A characterization of uniformly bounded cosine functions generators (Q917000)

Given a linear operator A on a Banach space X, a vector \(x\in X\) is called exponential, if \(x\in D^{\infty}(A)\) and \(\limsup_{n\to \infty}\| A^ nx\|^{1/n}<\infty.\) The following theorem is presented: Theorem 1. For the closed operator A, the following conditions are equivalent: 1. A is the generator of a uniformly bounded cosine family; 2. A is the generator of a bounded holomorphic semigroup in Re z\(>0\) (i.e. of angle \(\pi\) /2) and the set of its exponential vectors is dense in X. Unfortunately, the proof of the implication 2.\(\to 1\). is not correct. The reviewer's opinion is that the theorem should be modified - in this form it is not true. The paper contains also a sufficient condition for a cosine function \(C_ t\) to have the representation \(C_ t=(T_ t+T_{-t}),\) with a \(C_ 0\)-group \(T_ t\), \(t\in {\mathbb{R}}\). (Theorem 2.)